On the stability of Mindlin-Timoshenko plates

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Universität Konstanz

On the stability of Mindlin-Timoshenko plates

Hugo D. Fernández Sare

Konstanzer Schriften in Mathematik und Informatik Nr. 237, Oktober 2007

ISSN 1430-3558

Fach D 188, 78457 Konstanz, Germany E-Mail: preprints@informatik.uni-konstanz.de

WWW: http://www.informatik.uni-konstanz.de/Schriften/

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/3826/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-38264

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HUGO D. FERN ´ANDEZ SARE

Abstract. We consider a Mindlin-Timoshenko model with frictional dissipations acting on the equations for the rotation angles. We prove that this system is not exponentially stable independent of any relations between the constants of the system, which is different from the analogous’ one-dimensional case. Moreover, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.

1. Introduction

The conservative Mindlin-Timoshenko model in two dimensional case is given by ρhwtt−K

· ∂

∂x µ

ψ+∂w

∂x

¶ + ∂

∂y µ

ϕ+∂w

∂y

¶¸

= 0 (1.1)

ρh³

12 ψtt−D µ∂²ψ

∂x² +1−µ 2

∂²ψ

∂y² +1 +µ 2

∂²ϕ

∂x∂y

¶ +K

µ ψ+∂w

∂x

¶

= 0 (1.2) ρh³

12 ϕtt−D µ∂²ϕ

∂y² +1−µ 2

∂²ϕ

∂x² +1 +µ 2

∂²ψ

∂x∂y

¶ +K

µ ϕ+∂w

∂y

¶

= 0 (1.3) where Ω⊂R²is bounded,ρis the (constant) mass per unit of surface area,his the (uniform) plate thickness, µis Poisson’s ratio (0 < µ < ¹₂ in physical situations), D is the modulus of flexural rigidity and K is the shear modulus. The functions w, ψ and ϕ depend on (t, x, y)∈ [0,∞)×Ω, where w models the transverse displacement of the plate, and ψ, ϕ are the rotation angles of a filament of the plate, cp. [7, 8].

The main difference of this system to the analogous one-dimensional case (ϕ≡0) is that here another equation for rotation angles is considered. Note also that the coupling between the equations of the rotational angles (ψ, ϕ) and the displacement equation w, is weaker than in one dimension. Therefore the questions how it is possible to stabilize the system and to find ”sufficient” dissipations to produce exponential stability are interesting and thus is not much studied in the literature.

In [7], Lagnese considered a bounded domain Ω having a Lipschitz boundary Γ such that Γ = Γ0∪Γ1, where Γ0 and Γ1 are relatively open, disjoints subsets of Γ with Γ1 6=∅. He considered the following boundary conditions

2000Mathematics Subject Classification. 35 B 40, 74 H 40.

Key words and phrases. Timoshenko plates, non-exponential stability, polynomial stabilitiy.

The author was supported by the DFG-project “Hyperbolic Thermoelasticity” (RA 504/3-3).

1

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w=ψ=ϕ = 0 in Γ0 (1.4) K

µ∂w

∂x +ψ,∂w

∂y +ϕ

¶

·ν = m1 in Γ1 (1.5)

D µ∂ψ

∂x +µ∂ϕ

∂y , 1−µ 2

µ∂ϕ

∂x +∂ψ

∂y

¶¶

·ν = m2 in Γ1 (1.6)

D

µ1−µ 2

µ∂ϕ

∂x +∂ψ

∂y

¶ , ∂ϕ

∂y +µ∂ψ

∂x

¶

·ν = m3 in Γ1, (1.7) where ν := (ν1, ν2) is the unit exterior normal to Γ =∂Ω and {m1, m2, m3} are the linear boundary feedbacks given by

{m1, m2, m3}=−F{wt, ψt, ϕt}

with F = [fij] a 3×3 matrix of real L^∞(Γ1) functions such that F is symmetric and positive semidefinite on Γ1. In that conditions, Lagnese proved that the system (1.1)–(1.3) is exponentially stable, without any restrictions on the coefficients of the system. The same result is obtained by M. Rivera and P. Oquendo [10], where they consider in (1.5)–(1.7) boundary dissipations of memory type, that is

w+K Z _t

0

g1(t−s) µ∂w

∂x(s) +ψ(s),∂w

∂y(s) +ϕ(s)

¶

·νds = 0 in Γ1

ψ+D Z _t

0

g2(t−s)

µ∂ψ(s)

∂x +µ∂ϕ(s)

∂y , 1−µ 2

µ∂ϕ(s)

∂x +∂ψ(s)

∂y

¶¶

·νds = 0 in Γ1

ϕ+D Z _t

0

g3(t−s) µ1−µ

2

µ∂ϕ(s)

∂x +∂ψ(s)

∂y

¶

, ∂ϕ(s)

∂y +µ∂ψ(s)

∂x

¶

·νds = 0 in Γ1. With these boundary feedbacks, together with condition (1.4), they proved that the solutions of the system (1.1)–(1.3) are exponentially stable provided that the kernels have exponential behavior, and are polynomially stable for kernels of polynomial type. Similar dissipations are used by L. Santos [3], where the author considered a Timoshenko model in Ω⊂Rⁿ.

In this work, we are interested in introducing another type of dissipation. For example, taking into account the papers mentioned above, if we consider three frictional internal dissipations into the system, this is, introducing the termswtin (1.1),ψtin (1.2) andϕtin (1.3), the exponential behavior of the solutions of the system is easily obtained. The natural questions are the following: what happens if we remove one of these dissipations? and, of course, which is the ”natural” candidate to be removed?. Looking to the one dimensional model we can deduce some conclusions in order to solve these questions. In [12], M. Rivera and Racke considered the one dimensional damped Timoshenko system

ρ1ϕtt−k(ϕx+ψ)x = 0 in (0, L) (1.8) ρ2ψtt−bψxx+k(ϕx+ψ) +dψt = 0 in (0, L), (1.9) and proved that the solution of the system is exponentially stable if and only if the wave speeds of the system are equal, that is if

ρ1

k =ρ2

b . (1.10)

That Timoshenko model (1.8)-(1.9) with several type of dissipations has been studied by many authors, see for example [4, 5, 6, 11, 12, 14] and the references therein. The common

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point in almost every work is the condition (1.10), necessary and sufficient to obtain exponential stability. Note that, removing the dissipative term dψt in (1.9) and puttingϕt

in (1.8), we can deduce that the system is not exponentially stable independently if (1.10) holds or not. From these observations for the system (1.8)–(1.9) we can establish an equivalent problem in two dimensions. This is, introducing frictional dissipations in the rotational angle equations (1.2)-(1.3), we obtain a new dissipative system where the rate of decay for the solutions of that system appears as an open problem to be analyzed.

In other words, the purpose of this paper is to study the stability of the Timoshenko system

ρ1wtt−Kdiv µ

ψ+∂w

∂x, ϕ+∂w

∂y

¶

= 0 (1.11)

ρ2ψtt−Ddiv µ∂ψ

∂x +µ∂ϕ

∂y,1−µ 2

µ∂ϕ

∂x +∂ψ

∂y

¶¶

+K

· ψ+∂w

∂x

¸

+d1ψt= 0 (1.12) ρ2ϕtt−Ddiv

µ1−µ 2

µ∂ϕ

∂x +∂ψ

∂y

¶ ,∂ϕ

∂y +µ∂ψ

∂x

¶ +K

· ϕ+∂w

∂y

¸

+d2ϕt= 0 (1.13) where ρ1 :=ρh, andρ2 := ^ρh₁₂³ in the system (1.1)–(1.3). We will prove that the system (1.11)–(1.13) is not exponentially stable, independent of any relation between the constants of the system, which is a different result from the one obtained in [12] in the one-dimensional case, where the condition (1.10) was sufficient and necessary to obtain exponential stability.

Moreover, using multiplier techniques and Pr¨uss’ result [13], we will prove that the system (1.11)–(1.13) is polynomially stable with rates that can be improved depending on the initial data. We would like to add here that this is the first time that the asymptotic behavior of the system (1.11)–(1.13) is studied, and that our analysis shows the differences in the dimensions clearly.

The paper is organized as follows: In Section 2 we shall look to the existence and uniqueness results using semigroup theory. The Timoshenko system (1.11)–(1.13) is shown to be not exponentially stable subject to mixed boundary conditions in Section 3. Finally, in Sec- tion 4 we study the polynomial stability of the system (1.11)–(1.13) with Dirichlet boundary conditions.

2. Existence and uniqueness

We will use the standard notation H^k(Ω) or H₀^k(Ω) to denote usual Sobolev spaces of order kover the regular domain Ω, and set L²(Ω) =H⁰(Ω). We consider the Timoshenko system (1.11)–(1.13) with the following Dirichlet boundary conditions

w(x, t) = ψ(x, t) = ϕ(x, t) = 0 in ∂Ω×R⁺, (2.1) and initial conditions

w(x,0) =w0(x), wt(x,0) =w1(x), in Ω

ψ(x,0) =ψ0(x), ψt(x,0) =ψ1(x), in Ω (2.2) ϕ(x,0) =ϕ0(x), ϕt(x,0) =ϕ1(x), in Ω.

In order to obtain existence, uniqueness and stability results we will use semigroup theory.

For this purpose we rewrite the system as evolution equation for U = (w, wt, ψ, ψt, ϕ, ϕt)⁰ ≡ (u¹, u², u³, u⁴, u⁵, u⁶,)⁰.

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ThenU formally satisfies

Ut=A1U, U(0) =U0

whereU0= (w0, w1, ψ0, ψ1, ϕ0, ϕ1)⁰, andA1 is the (yet formal) differential operator

A1:=







0 Id 0 0 0 0

K

ρ1∆ 0 _ρ^K

1∂x 0 _ρ^K

1∂y 0

0 0 0 Id 0 0

−_ρ^K

2∂x 0 B1 −^d_ρ¹

2Id _ρ^D

2

¡_1+µ

2

¢∂_xy² 0

0 0 0 0 0 Id

−_ρ^K

2∂y 0 _ρ^D

2

¡_1+µ

2

¢∂_xy² 0 B2 −^d_ρ²

2Id







, (2.3)

where the differential operatorsBi(i= 1,2), are defined by B1 = D

ρ2

·

∂_x²+ µ1−µ

2

¶

∂_y²

¸

− k ρ2Id B2 = D

ρ2

·µ1−µ 2

¶

∂_x²+∂_y²

¸

− k ρ2Id.

Let

H1:=H₀¹(Ω)×L²(Ω)×H₀¹(Ω)×L²(Ω)×H₀¹(Ω)×L²(Ω)

be the Hilbert space. In order to endow the spaceH1with a norm associated to the energy of the system (1.11)–(1.13), we will use the following result.

Lemma 2.1. There existsα0>0 such that, for all(ψ, ϕ)∈[H₀¹(Ω)]², Z

Ω

h

|ψx|²+|ϕy|²+ µ1−µ

2

¶

|ψy+ϕx|²+µψxϕ_y+µϕyψ_x i

dxdy ≥ α£

||ψ||²_H1+||ϕ||²_H1

¤. Moreover, for every K0 >0 there exists β :=β(K0)>0 such that for all K ≥K0 and for all(w, ψ, ϕ)∈[H₀¹(Ω)]³,

Z

Ω

h

|ψ_x|²+|ϕ_y|²+ µ1−µ

2

¶

|ψ_y+ϕ_x|²+µψ_xϕ_y+µϕ_yψ_xi dxdy +K

Z

Ω

|ψ+wx|²dxdy+K Z

Ω

|ϕ+wx|²dxdy≥β£

||∇ψ||²_L2+||∇ϕ||²_L2+||∇w||²_L2

¤.

Proof. It’s a direct consequence of Korn’s Inequality, see [7]. ¤ Then, using the previous Lemma, we can obtain that

||U||²_H₁ = ||(u¹, u², u³, u⁴, u⁵, u⁶)||²_H₁

= ρ1||u²||²_L2+ρ2||u⁴||²_L2+ρ2||u⁶||²_L2+D||u³_x||²_L2+D||u⁵_y||²_L2

+K||u³+u¹_x||²_L2+K||u⁵+u¹_y||²_L2+µ(u³_x, u⁵_y)L²+µ(u⁵_y, u³_x)L² (2.4) is equivalent with the usual norm inH1.

Therefore, it is not difficult to prove that the operatorA1 is maximal – dissipative, that isA1 is the infinitesimal generator of aC0contraction semigroup onH1. Thus, we have the following result about existence and uniqueness of solutions.

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Theorem 2.2. Let U0= (w0, w1, ψ0, ψ1, ϕ0, ϕ1)⁰ ∈ H1. Then there exists a unique solution U(t) = (w, wt, ψ, ψt, ϕ, ϕt)⁰to system (1.11)–(1.13) with Dirichlet boundary conditions (2.1) satisfying

U ∈C(R⁺;D(A1))∩C¹(R⁺;H1).

Moreover, if U0∈D(Aⁿ₁), then

U ∈C^n−k(R⁺;D(A^k₁)) , k= 0,1,· · ·, n.

Remark 2.3. The same analysis can be applied to obtain existence and uniqueness results for mixed boundary conditions.

3. Non-exponential stability

In this Section we will prove that system (1.11)-(1.13) is not exponentially stable for suitable boundary conditions. In fact, we consider Ω⊂R² as the rectangle

Ω := [0, L1]×[0, L2], with L1, L2>0.

We define the sets

Γ1 :=

n

(x, y) : 0< x < L1, y= 0, L2

o Γ2 :=

n

(x, y) : 0< y < L2, x= 0, L1

o .

Note that Γ :=∂Ω = Γ1∪Γ2. The boundary conditions considered for the system (1.11)- (1.13) are the following

w= 0 in Γ ψ= 0 ,

µ1−µ 2

µ∂ϕ

∂x +∂ψ

∂y

¶ , ∂ϕ

∂y +µ∂ψ

∂x

¶

·ν= 0 in Γ1 (3.1) ϕ= 0 ,

µ∂ψ

∂x +µ∂ϕ

∂y , 1−µ 2

µ∂ϕ

∂x +∂ψ

∂y

¶¶

·ν = 0 in Γ2

where ν := (ν1, ν2) is the unit exterior normal to Γ = ∂Ω. Therefore, the semigroup formulation is given in the Hilbert space,

H2:=H₀¹(Ω)×L²(Ω)×H_Γ¹₁(Ω)×L²(Ω)×H_Γ¹₂(Ω)×L²(Ω), where

H_Γ¹_i(Ω) := ©

u∈H¹(Ω) : u= 0 in Γi

ª (i= 1,2) and with the same norm given by (2.4).

We shall use the following well-known result from semigroup theory (see e.g. [9, Theorem 1.3.2]).

Lemma 3.1. A semigroup of contractions {e^tA}t≥0 in a Hilbert space with norm k · k is exponentially stable if and only if

(i) the resolvent set%(A)ofA contains the imaginary axis and

(ii) lim sup

λ→±∞k(iλId− A)⁻¹k<∞ hold.

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Hence it suffices to show the existence of sequences (λn)n⊂Rwith

n→∞lim |λn|=∞, and (Un)n⊂D(A1), (Fn)n⊂ H, such that

(iλnId− A1)Un=Fn is bounded and lim

n→∞kUnkH1=∞.

AsFn≡F we choose F := (0, f²,0, f⁴,0, f⁶)⁰ with

f² := F²sin(δλ1x) sin(δλ2y), F²6= 0 (constant) f⁴ := F⁴cos(δλ1x) sin(δλ2y), F⁴6= 0 (constant) f⁶ := F⁶sin(δλ1x) cos(δλ2y), F⁶6= 0 (constant), where

λ1≡λ1,n:= nπ

δL1 , λ2≡λ2,n:= nπ

δL2 (n∈N), δ:=

rρ1

k. Finally we deffine

λ≡λn:=

q

λ²₁+λ²₂. (3.2)

The solutionU = (v¹, v², v³, v⁴, v⁵, v⁶)⁰ of the resolvent equation (iλId− A1)U =F

should satisfy

iλv¹−v² = 0 iλv²− k

ρ1(v³+v_x¹)x− −k

ρ1(v⁵+v¹_y)y = f² iλv³−v⁴ = 0 iλv⁴−D

ρ2

· v_xx³ +

µ1−µ 2

¶ v_yy³ +

µ1 +µ 2

¶ v_xy⁵

¸ + k

ρ2

(v³+v¹_x) +d1

ρ2

v⁴ = f⁴ iλv⁵−v⁶ = 0 iλv⁶− D

ρ2

·µ1−µ 2

¶

v⁵_xx+v_yy⁵ +

µ1 +µ 2

¶ v_xy³

¸ + k

ρ2

(v⁵+v_y¹) +d2

ρ2

v⁶ = f⁶.

(3.3)

Eliminatingv², v⁴, v⁶we obtain for v¹, v³, v⁵the following system

−λ²ρ1v¹−k(v³+v¹_x)x−k(v⁵+v¹_y)y = ρ1f²

−λ²ρ2v³−D£

v³_xx+¡_1−µ

2

¢v³_yy+¡_1+µ

2

¢v_xy⁵ ¤

+k(v³+v¹_x) +iλd1v³ = ρ2f⁴

−λ²ρ2v⁵−D£¡_1−µ

2

¢v_xx⁵ +v_yy⁵ +¡_1+µ

2

¢v_xy³ ¤

+k(v⁵+v¹_y) +iλd2v⁵ = ρ2f⁶.

(3.4)

System (3.4) can be solved by

v¹(x, y) := Asin(δλ1x) sin(δλ2y) v³(x, y) := Bcos(δλ1x) sin(δλ2y) v⁵(x, y) := Csin(δλ1x) cos(δλ2y)

whereA, B,C depend onλand will be determined explicitly in the sequel. Note that this choice is just compatible with the boundary conditions (3.1). System (3.4) is equivalent to

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findingA, B, C such that

−λ²ρ1A+kδ²¡

λ²₁+λ²₂¢

A+kδλ1B+kδλ2C = ρ1F² (3.5)

−λ²ρ2B+Dδ²λ²₁B+D µ1−µ

2

¶

δ²λ²₂B+D

µ1 +µ 2

¶

δ²λ1λ2C+

+kB+kδλ1A+iλd1B = ρ2F⁴ (3.6)

−λ²ρ2C+D

µ1−µ 2

¶

δ²λ²₁C+Dδ²λ²₂C+D

µ1 +µ 2

¶

δ²λ1λ2B+

+kC+kδλ2A+iλd2C = ρ2F⁶. (3.7) Using the definitions ofδandλ, we obtain from (3.5) that

B=−λ2

λ1

C+ 1 λ1

δF² (3.8)

or

C=−λ1

λ2B+ 1

λ2δF². (3.9)

Using (3.9) in (3.6) results

·

−λ² µ

ρ2−Dδ²

µ1−µ 2

¶¶

+k+iλd1

¸

B+kδλ1A=ρ2F⁴−D

µ1 +µ 2

¶

δ³λ1F². (3.10) Similarly, using (3.8) in (3.7) results

·

−λ² µ

ρ2−Dδ² µ1−µ

2

¶¶

+k+iλd2

¸

C+kδλ2A=ρ2F⁶−D

µ1 +µ 2

¶

δ³λ2F². (3.11) Let

Θ := ρ2−Dρ1

k

µ1−µ 2

¶

, (3.12)

then, using the definition ofδ, we obtain from (3.10)-(3.11) thatA, B satisfies

¡−λ²Θ +k+iλd1

¢B+kδλ1A = ρ2F⁴−D

µ1 +µ 2

¶

δ³λ1F² (3.13)

¢C+kδλ2A = ρ2F⁶−D

µ1 +µ 2

¶

δ³λ2F². (3.14) Remark 3.2. Note that the condition Θ = 0 in (3.12) gives a relationship (in 2-dimensional case) similar to the relation

ρ1

k =ρ2

b ,

which is necessary and sufficient condition for exponential stability in 1-dimensional case, see [12]. We will show that the system (1.11)-(1.13) is non-exponentially stable, independent of any relation between the coefficients of the system, in particular of Θ = 0 in (3.12).

Using (3.9) into (3.14) results

¢λ²₁

λ²₂B−kδλ1A = −λ1

λ2ρ2F⁶+D

µ1 +µ 2

¶ δ³λ1F² +¡

−λ²Θ +k+iλd2

¢δλ1F², (3.15)

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then, adding the equalities (3.13) and (3.15) yields

·¡

¢+¡

¢λ²₁ λ²₂

¸

B = ρ2F⁴−λ1

λ2

ρ2F⁶ +¡

¢δλ1F²,

this is

B =

ρ2F⁴−λ1

λ2ρ2F⁶+¡

¢δλ1F²

−λ²Θ µ

1 + λ²₁ λ²₂

¶ +k

µ 1 + λ²₁

λ²₂

¶ +iλ

µ d1+d2

λ²₁ λ²₂

¶ (3.16)

and using (3.13) we have A = ρ2

kδλ1F⁴−D

µ1 +µ 2

¶δ² kF²−¡

¢B (3.17)

withB given by (3.16). We define Q(λ) := −λ²Θ

µ 1 + λ²₁

λ²₂

¶ +k

µ 1 + λ²₁

λ²₂

¶ +iλ

µ

d1+d2λ²₁ λ²₂

¶ ,

where, using the definitions ofλi(i= 1,2), we can conclude that L := λ1

λ₂ = L2

L₁ > 0. (3.18)

ThereforeQ(λ) is given by Q(λ) = −λ²Θ¡

1 +L²¢ +k¡

1 +L²¢ +iλ¡

d1+d2L²¢

. (3.19)

We also define the following functions A1(λ) := ρ2

kδλ1F⁴+ 1 Q(λ)

· kρ2

¡LF⁶−F⁴¢

−D µ1 +µ

2

¶ δ²¡

1 +L²¢¸

(3.20)

A2(λ) := 1 Q(λ)

"

−λ⁴λ1Θ²δF²+iλ³λ1Θ (d1−d2)δF²+λ²λ1(d2d1+ 2Θk)δF²

+λ²Θ³

(F⁴−LF⁶)ρ2+D

µ1 +µ 2

¶δ²

k(1 +L²)´

−iλλ1(d1+d2)kδF²

+iλ

³

d1ρ2(LF⁶−F⁴)−D

µ1 +µ 2

¶

(d1+d2L²)

´

−δλ1kF²

#

. (3.21)

Then we have in (3.17) that

A=A1(λ) +A2(λ).

Recalling that

v²=iλv¹=iλAsin(δλ1x) sin(δλ2y) we get

v²=

³

iλA1(λ) +iλA2(λ)

´

sin(δλ1x) sin(δλ2y).

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Note that

||Un||H ≥ ||v²||L²

= ³ Z

Ω

|v²|²dxdy´_1/2

≥ −C1|λA1(λ)|+C2|λA2(λ)|,

where Ci :=Ci(L1, L2)>0,i= 1,2. Then, to complete our result, it is sufficient to show that,

(i) The sequence{λA1(λ)}λ⊂R⁺ is bounded, and

(ii) |λA2(λ)| → ∞asλ→ ∞, independent of any relation between the constants of the system.

In fact, using the definitions ofλ,λ1 in (3.20) we obtain

λA1(λ) = ρ₂ kδ

r 1 + 1

L²F⁴+ kρ2

¡LF⁶−F⁴¢

−D

µ1 +µ 2

¶ δ²¡

1 +L²¢

−λΘ¡ 1 +L²¢

+k λ

¡1 +L²¢ +i¡

d1+d2L²¢. Then {λA1(λ)}λ is bounded, which completes the proof of item (i). On the other hand, note that item (ii) is obvious in the case Θ6= 0. When Θ = 0 we have in (3.21) that

λA2(λ) = 1 Q0(λ)

"

2λ³λ1d2d1δF²−iλ²λ1(d1+d2)kδF²

+iλ²

³

d1ρ2(LF⁶−F⁴)−D µ1 +µ

2

¶

(d1+d2L²)

´

−δλλ1kF²

# , with

Q0(λ) = k(1 +L²) +iλ(d1+d2L²).

Therefore|λA2(λ)| −→ ∞. Thus we have proved

Theorem 3.3. The Timoshenko system (1.11)-(1.13) with boundary conditions (3.1) is not exponentially stable, independent of any relation between the constants of the system.

Remark 3.4. As in the 1-dimensional case, the non-exponential stability to Dirichlet boundary conditions (2.1) is still an open problem. Note also that the function that generates the non-exponential stability, that is {λA2(λ)}λ, has the behavior as|λA2(λ)| ∼ ◦(λ³), which produce the expectation that, to show polynomial stability results, we will need energies of higher order, see Section 4.

4. Polynomial stability

In this section we shall prove that the system (1.11)-(1.13) with boundary conditions (2.1) is polynomially stable. The energy of first order associated to the system (1.11)-(1.13) is given by

E1(t) :=E1(t;w, ψ, ϕ) =1 2

Z

Ω

h

ρ1|wt|²+ρ2|ψt|²+ρ2|ϕt|²+K|ψ+wx|²+K|ϕ+wx|²+D|ψx|² +D|ϕy|²+

µ1−µ 2

¶

D|ψy+ϕx|²+ 2Dµψxϕy

i

dxdy, (4.1)

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which is obtained multiplying the equations (1.11) by wt, (1.12) by ψt and (1.13) by ϕt. Also, we can define the energies

Ei+1(t) :=E1(t;∂_t⁽ⁱ⁾w, ∂_t⁽ⁱ⁾ψ, ∂_t⁽ⁱ⁾ϕ), i= 1,2,3. (4.2) It is not difficult to show that

d

dtEi(t) = −d1

Z

Ω

|∂_t⁽ⁱ⁾ψ|²dxdy−d2

Z

Ω

|∂_t⁽ⁱ⁾ϕ|²dxdy , i= 1,2,3,4. (4.3) We define

F1(t) :=

Z

Ω

[ρ1wtw+ρ2ψtψ+ρ2ϕtϕ+d1

2ψ²+d2

2 ϕ²]dxdy, (4.4) then, multiplying the equation (1.11) byw, (1.12) byψand (1.13) byϕ, results in

d

dtF1(t) = −D Z

Ω

h

|ψx|²+|ϕy|²+

µ1−µ 2

¶

|ψy+ϕx|²+ 2µψyϕx

i dxdy

−K Z

Ω

|ψ+wx|²dxdy−K Z

Ω

|ϕ+wx|²dxdy+ρ2

Z

Ω

|ψt|²dxdy +ρ2

Z

Ω

|ϕt|²dxdy+ρ1

Z

Ω

|wt|²dxdy. (4.5)

Letq: Ω→Rdefined byq(x, y) =x. We define F₂(t) := −D

Z

Ω

µ

ψ_xt+µϕ_yt, µ1−µ

2

¶

(ψ_yt−ϕ_xt)

¶

.∇wq(x, y)dxdy, (4.6) then, differentiating the equation (1.12) with respect to t and multiplying by q(x, y)wt in L²(Ω) results in

d

dtF2(t) = −K Z

Ω

|wt|²dxdy+ρ2

Z

Ω

ψtttq(x, y)wtdxdy+K Z

Ω

ψtq(x, y)wtdxdy +d1

Z

Ω

ψttq(x, y)wtdxdy+D Z

Ω

(ψxt+µϕyt)wtdxdy

−D Z

Ω

µ

ψxtt+µϕytt, µ1−µ

2

¶

(ψytt−ϕxtt)

¶

.∇wq(x, y)dxdy, where we can conclude that there exists

Ci:=Ci(ρ1, ρ2, K, D, µ,Ω)>0, i= 1,2, (4.7) such that

d

dtF2(t) ≤ −K 2

Z

Ω

|wt|²dxdy+C1

£||ψt||²_H1+||ϕt||²_H1

¤+C2

Z

Ω

£|ψtt|²+|ψttt|²¤ dxdy

−D Z

Ω

µ

2

¶

(ψytt−ϕxtt)

¶

.∇wq(x, y)dxdy. (4.8) We will use the letterC to denote several positive constants defined as in (4.7). Defining

F3(t) := F1(t) +4ρ1

K F2(t), (4.9)

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and using (4.5) and (4.8) we have d

dtF3(t) ≤ −2E1(t) +C£

||ψt||²_H1+||ϕt||²_H1

¤+C Z

Ω

£|ψtt|²+|ψttt|²¤ dxdy

−4ρ1D K

Z

Ω

µ

ψxtt+µϕytt,

µ1−µ 2

¶

(ψytt−ϕxtt)

¶

.∇wq(x, y)dxdy.(4.10) Remark 4.1. In (4.10) we have already the first order energy with negative sign, but it is necessary to estimate other higher-order terms. The following functionals will be defined in order to estimate these terms.

First, note that using Korn’s Inequality [2], we have that there exists constantsα, β >0 such that (see [7])

Z

Ω

h

|ψx|²+|ϕy|²+ µ1−µ

2

¶

|ψy+ϕx|²+ 2µψxϕy

i

dxdy ≥ α£

||ψ||²_H1+||ϕ||²_H1

¤ (4.11)

and Z

Ω

h

|ψx|²+|ϕy|²+ µ1−µ

2

¶

|ψy+ϕx|²+ 2µψxϕy

i dxdy +K

Z

Ω

|ψ+wx|²dxdy+K Z

Ω

|ϕ+wx|²dxdy≥β£

||∇ψ||²_L2+||∇ϕ||²_L2+||∇w||²_L2

¤. (4.12)

On the other hand, differentiating the equations (1.12)-(1.13) with respect tot and multiplying byψtandϕtrespectively, results in

d dt

Z

Ω

ρ2[ψttψt+ϕttϕt]dxdy=−D Z

Ω

h

|ψxt|²+|ϕyt|²+ µ1−µ

2

¶

|ψyt+ϕxt|²+2µψytϕxt

i dxdy +ρ2

Z

Ω

|ψtt|²dxdy−K Z

Ω

(ψt+wxt)ψtdxdy−d1

Z

Ω

ψttψtdxdy +ρ2

Z

Ω

|ϕtt|²dxdy−K Z

Ω

(ϕt+wxt)ϕtdxdy−d2

Z

Ω

ϕttϕtdxdy. (4.13) Then, defining

F4(t) :=

Z

Ω

[ρ2ψttψt+ρ2ϕttϕt+K∇w.(ψt, ϕt)] dxdy (4.14) and using (4.11) we obtain

d

dtF4(t) ≤ −Dα£

||ψt||²_H1+||ϕt||²_H1

¤+C Z

Ω

£|ψtt|²+|ϕtt|²¤ dxdy +K

Z

Ω

(wxψtt+wyϕtt) dxdy. (4.15) Let

F5(t) := F3(t) + C

DαF4(t), (4.16)

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then, from (4.10) and (4.15) we have d

dtF5(t) ≤ −2E1(t) +C Z

Ω

£|ψtt|²+|ϕtt|²+|ψttt|²¤

dxdy+CK Dα

Z

Ω

(wxψtt+wyϕtt) dxdy

−4ρ1D K

Z

Ω

µ

2

¶

(ψytt−ϕxtt)

¶

.∇wq(x, y)dxdy. (4.17) Note that, using the definition ofE1(t) and the inequality (4.12), we obtain

−2E1(t) ≤ −E1(t)−β

2||∇w||²_L2. (4.18) Therefore, applying (4.18) in (4.17) we can deduce that

d

dtF5(t) ≤ −E1(t)−β

4||∇w||²_L2+C Z

Ω

|ψttt|²dxdy +Cβ

£||ψtt||²_H1+||ϕtt||²_H1

¤, (4.19)

whereCβ>0 is defined as (4.7) and depends also ofβ >0.

Similarly as in (4.13), differentiating equations (1.12)-(1.13) with respect tottwo times, and multiplying byψtt andϕtt respectively, we can deduce

d dt

Z

Ω

ρ2[ψtttψtt+ϕtttϕtt]dxdy≤ −Dα£

||ψtt||²_H1+||ϕtt||²_H1

¤+ρ2

Z

Ω

£|ψttt|²+|ϕttt|²¤ dxdy

−K Z

Ω

(ψtt+wxtt)ψttdxdy−d1

Z

Ω

ψtttψttdxdy

−K Z

Ω

(ϕ_tt+w_xtt)ϕ_ttdxdy−d₂ Z

Ω

ϕ_tttϕ_ttdxdy, (4.20) where inequality (4.11) is used. We define

F6(t) :=

Z

Ω

[ρ2ψtttψtt+ρ2ϕtttϕtt−K∇wt.(ψtt, ϕtt) +K∇w.(ψttt, ϕttt)] dxdy. (4.21) Then, from (4.20) we deduce

d

dtF6(t) ≤ −Dα£

||ψtt||²_H1+||ϕtt||²_H1

¤+C Z

Ω

−K Z

Ω

∇w.(ψtttt, ϕtttt)dxdy (4.22) Finally we define

F7(t) := F5(t) + Cβ

DαF6(t). (4.23)

Then, from (4.19) and (4.22) results in d

dtF7(t) ≤ −E1(t)−β

4||∇w||²_L2+C Z

Ω

−K Z

Ω

∇w.(ψtttt, ϕtttt)dxdy, and we can deduce that

d

dtF7(t) ≤ −E1(t) +C0

Z

Ω

£|ψttt|²+|ϕttt|²+|ψtttt|²+|ϕtttt|²¤

dxdy, (4.24)

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where C0 >0 is a constant defined as (4.7), and also depends on the constants given by Korn’s Inequality.

Now we are in the position to prove the main result of this section.

Theorem 4.2. Suppose that the initial data verify

U0:= (w0, w1, ψ0, ψ1, ϕ0, ϕ1)⁰∈D(A⁴).

Then the first order energyE1(t)associated to system (1.11)-(1.13) with boundary conditions (2.1) decays polynomially to zero as time goes to infinity, that is, there exists a positive constant C, being independent of the initial data, such that

E1(t)≤C t

X4 i=1

Ei(0). (4.25)

Moreover, if U0∈D(A^4k), then

||T(t)U0||H ≤ C_k

t^k ||A^4kU0||H, ∀k= 1,2,3, ... (4.26) where{T(t)}t≥0 is the semigroup associated to system (1.11)-(1.13) with infinitesimal gen- eratorA defined as (2.3).

Proof. We defineL(t) as

L(t) := C0

d X4 i=1

E_i(t) +F₇(t),

whered:= min{d1, d2}>0, withd1, d2given by the system (1.11)-(1.13). Then, using (4.3) and (4.24) we obtain

d

dtL(t)≤ −E1(t).

Therefore Z _t

0

E1(s)ds≤ L(0)− L(t) , ∀t≥0. (4.27) On the other hand, it is not difficult to prove that there exists a constantC >0 such that

L(0)− L(t)≤C X4 i=1

Ei(0) , ∀t≥0. (4.28)

From (4.27)–(4.28) we obtain Z _t

0

E1(s)ds≤C X4 i=1

Ei(0). (4.29)

Then, since

d dt

n tE1(t)o

=E1(t) +td

dtE1(t)≤E1(t), from (4.29) we get

E1(t)≤C t

X4 i=1

Ei(0), which completes (4.25) and show that (4.26) holds, fork= 1.

Finally, ifU0∈D(A^4k),k≥2, we use Pr¨uss’ results [13] to obtain (4.26), which completes

the proof. ¤

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Department of Mathematics and Statistics, University of Konstanz, 78457, Konstanz, Germany E-mail address:hugo.fernandez-sare@uni-konstanz.de